We prove some multiplicity results for a class of singular quasilinear elliptic problems involving the critical Hardy-Sobolev exponent and singularities on a half-space.

In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567].

Abstract. We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R . We focus on the cases when f±(x, u) = ±(−λ|u| u+ |u|u), where m(x) := max{p1(x), p2(x)} < q(x) < N ·m(x) N−m(x) for any x ∈ Ω. In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove that if λ is...

In this article a new method for moving from local to global results in variable exponent function spaces is presented. Several applications of the method are also given: Sobolev and trace embeddings; variable Riesz potential estimates; and maximal function inequalities in Morrey spaces are derived for unbounded domains.

Available online xxxx Keywords: Nonlinear heat equation Blow up Sobolev spaces with variable exponents a b s t r a c t In this paper we consider a nonlinear heat equation with nonlinearities of variable-exponent type. We show that any solution with nontrivial initial datum blows up in finite time. We also give a two-dimension numerical example to illustrate our result.

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent λ = n−α (that is for the case of α > n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequ...

with ai(x),pi(x) ∈ C1( ), pi(x) > 1,ai(x)≥ 0. Basing on the weighted variable exponent Sobolev space, a new kind of weak solutions of the equation is introduced. Whether the usual Dirichlet homogeneous boundary value condition can be imposed depends on whether ai(x) is degenerate on the boundary or not. If some of {ai(x)} are degenerate on the boundary, a partial boundary value condition is imp...

We study the homogeneous Dirichlet problem for the equation ut = n ∑ i=1 Di ( ai|Di(|u|m(x)−1u)|pi(x,t)−2Di(|u|m(x)−1u) ) +b|u|σ(x,t)−2u with given exponents m(x) , pi(x,t) and σ(x,t) . It is proved that the problem has a solution in a suitable variable exponent Sobolev space. In dependence on the properties of the coefficient b and the exponents of nonlinearity, the solution exists globally or...